Problem 20 - Fundamentals of Mathematics

Question

Find each value. \(\left|\frac{-32}{8}-\frac{-15-5}{5}\right|\)

Step-by-Step Solution

Answer

The value is 0.

Step 1: Simplify Each Fraction

First, simplify each fraction separately. The first fraction is \(-\frac{32}{8}\). Dividing \(-32\) by 8 gives \(-4\). So, the first fraction simplifies to \(-4\).For the second fraction, \(-\frac{-15-5}{5}\), first simplify the numerator by performing the operation \(-15 - 5\), which results in \(-20\). Now divide \(-20\) by 5, which equals \(-4\). So, the second fraction simplifies to \(-4\) as well.

Step 2: Subtract the Simplified Fractions

Now subtract the two simplified fractions: \(-4 - (-4)\). This operation can be rewritten as \(-4 + 4\), because subtracting a negative is the same as adding the positive equivalent. Therefore, \(-4 + 4 = 0\).

Step 3: Apply the Absolute Value Function

The expression now becomes \(|0|\). The absolute value function of 0 is 0, because the absolute value of any number is its distance from zero on the number line.

Key Concepts

Fractions

Fractions are like slices of a pie, representing parts of a whole. Understanding fractions is key in many areas of math. A fraction consists of two numbers, the numerator and the denominator. The numerator is the top number, and it shows how many parts we're considering. The denominator is the bottom number, showing the total number of equal parts the whole is divided into.
For example, \(-\frac{32}{8}\) tells us about 32 parts where the whole is divided into 8 equal slices.

• Improper Fractions: When the numerator is larger than the denominator, like \(\frac{9}{4}\).
• Proper Fractions: When the numerator is smaller than the denominator, like \(\frac{1}{3}\).
• Mixed Numbers: A whole number and a proper fraction combined, like 2 \(\frac{1}{2}\).

In our exercise, both fractions, \(-\frac{32}{8}\) and \(-\frac{-15-5}{5}\), were treated to see how they represent parts of the respective numerators and denominators.

Simplification

Simplification is the process of making fractions easier to work with. It's like having a cluttered room and tidying it up. By simplifying, we make math problems more straightforward and efficient to solve. Simplifying involves reducing the fraction to its smallest form.

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by that GCF.

In the original exercise, simplifying \(-\frac{32}{8}\) led to \(-4\), thanks to dividing both -32 and 8 by their GCF, which is 8. Likewise, \(-\frac{-20}{5}\) is simplified by dividing -20 by 5, also leading to \(-4\).
Simplification helps us quickly see relationships in equations and solve them more effectively.

Numerator and Denominator

Think of the numerator and denominator as the heart and soul of a fraction. Their relationship shapes the fraction's value. The numerator, being the top number, represents how many parts are taken.
The denominator, being the bottom number, shows into how many equal parts the whole is divided.In the exercise, we examine two cases:

• Numerator: -32, Denominator: 8 in \(-\frac{32}{8}\). This means the fraction considers 32 parts out of 8, simplifying to -4 with whole parts balanced atop the base of 8.
• For \(-\frac{-15-5}{5}\), after calculating the numerator as (-15 - 5 = -20), you have a simple division step by 5.

Understanding how these two numbers interact helps enormously in both the comprehension and manipulation of fractions in any equation or function.